These can be a pain to deal with, because it means playing with multiple styles of bit. What is she in now? Chewing on the bit generally means she doesn't like something about it.
For example, a horse that throws a fit over a double jointed mouth with a roller is probably going to be much happier in a single joint or Mullen. Lots of horses hate 'busy' bits. If she's in a single joint, she may like the added lightness of a double joint, like a lozenge or half moon mouth.
Then there's height. Is she sucking the bit up to be chewing on it audibly? If her moving her jaw is creating that sound, her bit is too high. Take a look inside her mouth and see where her teeth all are in relation to her lips, and play with raising and lowering it a hole at a time, depending on where it is now.
Then, lastly, rider. Is she chewing at any certain point, or constantly? Constantly indicates something about equipment is bothering her, but if she does it only when she's cued or only when she circles one way, or when she's frustrated, then the bit probably isn't the issue. She may be protesting the contact, taking out her anxiety/frustration on the bit, or indicating that something about doing that thing that triggers the chomping bothers her. One horse I worked with would start tossing his head and chomping at the bit when he was asked to circle to the right, and it turned out he was developing arthritis in his right knee.
Melissa, checkout liberty bridles! I don't suggest showing in it but it does look a lot like a bit... They're open to trials... Might be something good to school in? I saw that someone said it helped their horse while being bitted from just schooling in it.
Mark, Wouldn't giving her something to play with on the bit be counter-productive, though? She is opening and closing her mouth mostly when we halt, but always is messing around with it. Doesn't seem to affect the contact much, though. Are Figure-8 bridles legal in Dressage?
Natalie, I have not tried a bitless, but have considered getting one to mess around in. However, I know they are not legal for showing...
Thanks for your help, everyone!
I would suggest some stretch work before you ask her to come a little more into a frame. Have her reach long and low with your leg but keep a very loose contact in the reins. She is most likely doing this because she is nervous or uncomfortable in the contact so try bringing it up slow after allowing her to adjust a bit.
What mark said. Also The bit could also be sitting too high or low in the horses mouth.
Maybe get a bit with a roller in the middle or copper? And I know you said she didn't like restriction but have you tried a figure 8 bridle?